hubba-
soundproofing the closet will be hard, because you will be working around existing walls. I would suggest adding a room in the corner of your living room. The key were will be to build a room within a room. This is called isolation. The idea is to have your room floating on air. Obviously, we can decouple (disconnect) all four walls and the ceiling from your existing structure, but the floor on the other hand is a problem. If you get sound vibrations into your floor you are pretty much screwed. This is why you need to isolate your booth. Most people float them on neoprene pucks or strips. This absorbs most of the vibrations your booth will give off when soundwaves hit the walls, and keep them out of the floor.
as for the room. there are some important things
1. Mass, Air, Mass......................When constructing your walls, keep in mind that you only want two layers of mass total on your wall. For example, two layers of drywall on top of each other counts as one layer of mass (also known as a leaf). Next would go the studs, which are not mass, so that empty space between counts as air. This is your air space that you can only have one of. It is very important for this air space to be as tight as possible, which means that there are no air gaps from one side of the wall to the other. After this airspace, there is another layer or two of drywall, counting as your next layer. This makes a MASS AIR MASS wall. Keep this rule in mind when designing your wall.
2. Airtight. As I said before, you will need to make it airtight as possible. Any where air can get thru so can sound. This means all electrical outlets and doors and windows must be sealed well with acoustical caulk. There should be no flanking of air into the next room.
3. Mass. On each leaf, you want as much mass as you can get. Most people use two or even three layers of drywall on each side. This will give you a hefty ammount of soundproofing, assuming everything else is done right. One important thing to keep in mind. Both of your layers should not be the same thickness. For example,, insted of using two layers of 1/2 inch drywall on both sides, mabye only do that on one side, and then a layer of 1/2 and 3/8 on the other. This will keep from resonant frequencies from getting thru the walls.
4. Windows and doors. You may not need a window, but will obviously need a door. Two layers of thick laminated safety glass, with an airspace between. Again, mass air mass on the window. The door needs to be sealed once it is shut, so look into a drop seal door kit. Basically, once the door is shut, it will put an airtight stopper in the little gap at the bottom. You also need to weatherstrip around the entire door, to ensure that nothing gets thru the crack. You need a good, solid core door, nice and thick and heavy. This will probably be your weakest link in the room.
5. Caulking. CAULK EVERYTHING. This includes electrical boxes, seams in the drywall, corners, where the drywall meets the floor, everything. It should be like a ziplock bag in there. Windows, door frames, etc, all need to have a generous ammount of acoustical rated caulk to them.
there a lot more to go, but this will give you an idea, post back with any more questions